EXPLICIT MODULI SPACES FOR CONGRUENCES OF ELLIPTIC CURVES

TOM FISHER

Abstract. We determine explicit birational models over ℚ for the modular surfaces parametrising pairs of N-congruent elliptic curves in all cases where this surface is an elliptic surface. In each case we also determine the rank of the Mordell-Weil lattice and the geometric Picard number.

1. Introduction

Let N ≥ 2 be an integer. A pair of elliptic curves are said to be N-congruent, if their N-torsion subgroups are isomorphic as Galois modules. Such an isomorphism raises the Weil pairing to the power ε for some ε ∈ (ℤ/Nℤ)^{×}. In this situation we say that the N-congruence has power ε. Since multiplication-by-m on one of the elliptic curves (for m an integer coprime to N) changes ε to m^{2}ε, we are only ever interested in the class of ε ∈ (ℤ/Nℤ)^{×} mod squares.

Let Z(N, ε) be the surface that parametrises pairs of N-congruent elliptic curves with power ε. This is a surface defined over ℚ. We only consider Z(N, ε) up to birational equivalence. Kani and Schanz [14, Theorem 4] classified the geometry of these surfaces, explicitly determining the pairs (N, ε) for which Z(N, ε) is birational over ℂ to either (i) a rational surface, (ii) an elliptic K3-surface, (iii) an elliptic surface with Kodaira dimension one (also known as a properly elliptic surface), or (iv) a surface of general type. In case (i) it is known that the surface is rational over ℚ. We show in cases (ii) and (iii) that the surface is birational over ℚ to an elliptic surface, determining in each case a Weierstrass equation for the generic fibre as an elliptic curve over ℚ(T). One application of these explicit equations is that we are then able to use the methods of van Luijk and Kloosterman to compute the geometric Picard number of each surface.

The problem of computing Z(N, ε) is closely related to that of computing the modular curves X_{E}(N, ε) parametrising the elliptic curves N-congruent (with power ε) to a given elliptic curve E. Equations for X_{E}(N, ε), and the family of curves it parametrises, have been determined as follows. The cases N ≤ 5 were treated by Rubin and Silverberg [21], [23], [25] for ε = 1, and by Fisher [7], [8] for ε ≠ 1. The case N = 7 was treated by Halberstadt and Kraus [12] for ε = 1, and by Poonen, Schaefer and Stoll [20] for ε ≠ 1. The case N = 8 was treated by Chen [6], and the cases N = 9 and N = 11 by Fisher [9], [10].

Date: 26th April 2018.

2010 Mathematics Subject Classification. 11G05, 11F80, 14J27.

Key words and phrases. elliptic curves, Galois representations, elliptic surfaces.

If N is not a prime power, then in principle we obtain equations for X_{E}(N, ε) as a fibre product of modular curves of smaller level. Equations that are substantially better than this have been obtained in the case (N, ε) = (6, 1) by Rubin and Silverberg [22], and independently Papadopoulos [19], and in the cases (N, ε) = (12, 1) and (12, 7) by Chen [5, Chapter 7]. Chen also gives equations for X_{E}(N, ε) in the cases (N, ε) = (6, 5) and (10, 1).

The equations for X_{E}(N, ε) do immediately give us equations for Z(N, ε), but unfortunately this does not always make it easy to find the elliptic fibrations. The main purpose of this note is to record the transformations that work in each case.

According to [14, Theorem 4] the surface Z(N, ε) is rational over ℂ for all N ≤ 5, and in the cases N = 6, 7, 8 with ε = 1. In each of these cases Z(N, ε) is rational over ℚ, as follows (see [5, Chapter 8]) from the results cited above.

In our terminology, it is part of the definition of an elliptic surface that it has a section. As we describe below, some of the cases in the next two theorems were already treated in [6], [10], [16].

Theorem 1.1. The surfaces Z(N, ε) that are birational over ℂ to an elliptic K3surface, are in fact birational over ℚ to an elliptic surface. The generic fibres are the elliptic curves over ℚ(T) with the following Weierstrass equations.

Z(6, 5) : y^{2} + 3T (T − 2)xy + 2(T − 1)(T + 2)^{2}(T^{3} − 2)y = x^{3} − 6(T − 1)(T^{3} − 2)x^{2},
Z(7, 3) : y^{2} = x^{3} + (4T^{4} + 4T^{3} − 51T^{2} − 2T − 50)x^{2} + (6T + 25)(52T^{2} − 4T + 25)x,
Z(8, 3) : y^{2} = x^{3} − (3T^{2} − 7)x^{2} − 4T^{2}(4T^{4} − 15)x + 4T^{2}(53T^{4} + 81T^{2} + 162),
Z(8, 5) : y^{2} = x^{3} − 2(T^{2} + 19)x^{2} − (4T^{2} − 49)(T^{4} − 6T^{2} + 25)x,
Z(9, 1) : y^{2} + (6T^{2} + 3T + 2)xy + T^{2}(T + 1)(4T^{3} + 9T + 9)y

= x^{3} − (16T^{4} + 12T^{3} + 9T^{2} + 6T + 1)x^{2},

Z(12, 1) : y^{2} + 2(5T^{2} + 9)xy + 96(T^{2} + 3)(T^{2} + 1)^{2}y = x^{3} + (T^{2} + 3)(11T^{2} + 1)x^{2}.

Theorem 1.2. The surfaces Z(N, ε) that are birational over ℂ to a properly elliptic surface, are in fact birational over ℚ to an elliptic surface. The generic fibres are the elliptic curves over ℚ(T) with the following Weierstrass equations.

Z(8, 7) : y^{2} = x^{3} + 2(4T^{6} − 15T^{4} + 14T^{2} − 1)x^{2} + (T^{2} − 1)^{4}(16T^{4} − 24T^{2} + 1)x,
Z(9, 2) : y^{2} + 3(4T^{3} + T^{2} − 2)xy + (T − 1)^{3}(T^{3} − 1)(4T^{3} − 3T − 7)y
= x^{3} − 3(T + 1)(T^{3} − 1)(9T^{2} + 2T + 1)x^{2},

Z(10, 1) : y^{2} − (3T − 2)(6T^{2} − 5T − 2)xy
− 4T^{2}(T − 1)^{2}(4T^{2} − 2T − 1)(27T^{3} − 54T^{2} + 16T + 12)y
= x^{3} + T^{2}(T − 1)(27T^{3} − 54T^{2} + 16T + 12)x^{2},
Z(10, 3) : y^{2} + (T^{3} − 8T^{2} − 9T − 8)xy + 2T^{2}(T^{3} − T^{2} − 3T − 3)(7T^{2} + 2T + 3)y
= x^{3} + 2(3T + 2)(T^{3} − T^{2} − 3T − 3)x^{2},
Z(11, 1) : y^{2} + (T^{3} + T)xy = x^{3} − (4T^{5} − 17T^{4} + 30T^{3} − 18T^{2} + 4)x^{2}
+ T^{2}(2T − 1)(3T^{2} − 7T + 5)^{2}x.

Although we have not made it formally part of the statements of Theorems 1.1 and 1.2, our methods do also give the moduli interpretations of these surfaces. In other words, given a point on one of these surfaces (away from a certain finite set of curves) we can determine the corresponding pair of N-congruent elliptic curves. The fact that N-congruent elliptic curves over ℚ have traces of Frobenius (at all primes of good reduction) that are congruent mod N, then provides some very useful check on our calculations.

The second part of the following corollary was conjectured by Kani and Schanz [14, Conjecture 5], and its proof (for ε = 1) was completed by Zexiang Chen in his PhD thesis [5]. For N sufficiently large it is expected (with variants of this conjecture variously attributed to Frey, Mazur, Kani and Darmon) that the conclusions of the corollary are false.

Corollary 1.3. Let N ≤ 12 and ε ∈ (ℤ/Nℤ)^{×} with ε = 1 if N = 11 or 12.

(i) There are infinitely many pairs of non-isogenous elliptic curves over ℚ(T) that are N-congruent with power ε.

(ii) There are infinitely many pairs of non-isogenous elliptic curves over ℚ that are N-congruent with power ε.

Moreover the j-invariants j_{1} and j_{2} of the elliptic curves in (i) (resp. (ii)) correspond to infinitely many curves (resp. points) in the (j_{1}, j_{2})-plane.

Proof. In Table 3 we list at least one ℚ-rational section of infinite order for each of the elliptic surfaces in Theorems 1.1 and 1.2. This proves the first part. The second part follows by specialising T. See the proof of [10, Theorem 1.5] for further details. The final sentence of the statement is included to guard against various “cheat” proofs, where new examples are generated from old by taking quadratic twists, or making substitutions for T. □

Remark 1.4. If elliptic curves E_{1} and E_{2} are N-congruent with power ε = −1, then the quotient of E_{1} × E_{2} by the graph of the N-congruence is a principally polarised abelian surface. The surface Z(N, −1) may then be interpreted as a Hilbert modular surface, parametrising degree N morphisms from a genus 2 curve to an elliptic curve. At the outset of our work, this moduli interpretation had not been made explicit for any N > 5. Remarkably however, this approach has been used by A. Kumar [16] to independently obtain results equivalent to the first two parts of Theorem 1.1 and the first three parts of Theorem 1.2. As far as we are aware, his methods do not generalise to ε ≠ −1.

In Tables 1 and 2 we record some further data concerning the elliptic surfaces in Theorems 1.1 and 1.2. Since a K3-surface may admit many elliptic fibrations, the data in Table 1 comes with the caveat that it relates to the elliptic fibration we happened to find in Theorem 1.1. Since a properly elliptic surface has a unique elliptic fibration, there is no such caveat for Table 2. We list in each case the Kodaira symbols of the singular fibres (with bracketing to indicate fibres that are Galois conjugates), the order of the torsion subgroup over ℚ(T), the ranks of the group of sections over ℚ(T) and \overline{ℚ}(T), and finally the geometric Picard number ρ. The lower bounds on the ranks are immediate from the independent sections of infinite order listed in Tables 3 and 4. The upper bounds on the ranks, and the geometric Picard numbers are justified in Section 4.

Table 1. The elliptic K3-surfaces in Theorem 1.1

(N, ε)  singular fibres  |tors| rank/ℚ rank/\overline{ℚ} ρ
(6, 5) (I_{2}, I_{2}), I_{3}, (I_{3}, I_{3}, I_{3}), I_{4}, I_{4}  1  2  2  20
(7, 3) I_{1}, I_{2}, (I_{2}, I_{2}), (I_{2}, I_{2}), I_{3}, I_{10}  2  2  2  20
(8, 3) (I_{1}, I_{1}), I_{2}, (I_{2}, I_{2}), (I_{2}, I_{2}), (I_{3}, I_{3}), I_{0}^{∗}  1  4  5  20
(8, 5) I_{2}, I_{2}, (I_{2}, I_{2}), (I_{2}, I_{2}), (I_{3}, I_{3}), I_{0}^{∗}  2  2  4  20
(9, 1) (I_{1}, I_{1}, I_{1}), I_{2}, (I_{2}, I_{2}, I_{2}), I_{3}, I_{4}, I_{0}^{∗}  1  3  4  19
(12, 1) (I_{1}, I_{1}, I_{1}, I_{1}, I_{1}, I_{1}, I_{1}, I_{1}), (I_{4}, I_{4}), (I_{4}, I_{4}) 1  3  5  19

Table 2. The properly elliptic surfaces in Theorem 1.2

(N, ε)  singular fibres  |tors| rank/ℚ rank/\overline{ℚ} ρ
(8, 7) I_{2}, (I_{2}, I_{2}), (I_{2}, I_{2}), (I_{3}, I_{3}), I_{4}, I_{8}, I_{8}  2  1  2  30
(9, 2) (I_{2}, I_{2}, I_{2}), (I_{3}, I_{3}), (I_{3}, I_{3}, I_{3}), I_{9}, I_{0}^{∗}  1  2  2  29
(10, 1) (I_{2}, I_{2}), (I_{2}, I_{2}), (I_{3}, I_{3}, I_{3}), I_{5}, I_{10}, IV  1  1  1  28
(10, 3) (I_{1}, I_{1}, I_{1}), I_{2}, (I_{2}, I_{2}), (I_{2}, I_{2}), (I_{3}, I_{3}, I_{3}), I_{4}, I_{4}, I_{6} 1  3  4  28
(11, 1) (I_{1}, I_{1}, I_{1}), I_{2}, (I_{2}, I_{2}, I_{2}), I_{3}, I_{4}, (I_{4}, I_{4}), I_{10}  2  2  2  28

Table 3. Mordell-Weil generators over ℚ(T)

(N, ε)  x-coordinates of independent sections of infinite order
(6, 5) 0, 2T^{4} − 4T,
(7, 3) 4T^{2} + 20T + 25, 6T + 25,
(8, 3) −7, −T^{2} + 9, −4T^{2} − 6T, (4T^{5} − 2T^{4} + 10T^{3} + 6T^{2} + 18T)/(T − 1)^{2},
(8, 5) −4T^{2} + 49, 2T^{3} + 19T^{2} + 60T + 63,
(9, 1) 0, 4T^{4} + 2T^{3} − 2T^{2}, 4T^{4} + 4T^{3} + 9T^{2} + 18T + 9,
(12, 1) 0, −12T^{4} − 24T^{2} − 12, 4T^{6} + 12T^{4} − 4T^{2} − 12,
(8, 7) 4T^{6} + 4T^{5} − 9T^{4} − 10T^{3} + 4T^{2} + 6T + 1,
(9, 2) 0, 2T^{5} − 8T^{3} + 4T^{2} + 6T − 4,
(10, 1) 0,
(10, 3) 0, 2T^{5} − 4T^{4} − 4T^{3} + 6T, 4T^{5} − 2T^{4} − 14T^{3} − 18T^{2} − 6T,
(11, 1) T^{4} + 4T^{2} + 4, 3T^{2} − 7T + 5.

Table 4. Additional Mordell-Weil generators over \overline{ℚ}(T)

(N, ε) d  section of infinite order defined over ℚ(\sqrt{d})
(8, 3) −2 (−2T^{4} − 5T^{2} − 9, (2T^{6} + 5T^{4} + 20T^{2} + 9)\sqrt{−2}),
(8, 5) −3 (−2T^{3} + T^{2} + 18T − 35, (12T^{3} − 6T^{2} − 108T + 210)\sqrt{−3}),
(8, 5) −1 (16T^{2} − 196, (8T^{4} − 346T^{2} + 3038)\sqrt{−1}),
(9, 1) −3 (−(19/3)T^{4} − 15T^{3} − 9T^{2}, …),
(12, 1) −3 (−12T^{4} − 40T^{2} − 12, …),
(12, 1) −1 (−16T^{4} − 64T^{2} − 48, …),
(8, 7) −3 (−4T^{6} − 20T^{5} − 39T^{4} − 36T^{3} − 14T^{2} + 1, …),
(10, 3) −3 (−7T^{6} − 23T^{5} − 30T^{4} − 15T^{3} − 9T^{2}, …).

We organise the proofs of Theorems 1.1 and 1.2 as follows. The cases N = 8 and N = 9 were already treated in [6], [10], by starting from equations for X_{E}(N, ε). In Section 2 we use a similar approach to treat the cases (N, ε) = (7, 3), (11, 1) and (12, 1). Then in Section 3 we treat the cases (N, ε) = (6, 5), (10, 1) and (10, 3) by exhibiting Z(N, ε) as a degree 3 cover of a K3-surface.

The calculations described in this paper were carried out using Magma [1]. Accompanying Magma files are available from the author’s website. We assume that the reader is familiar with the standard techniques for putting an elliptic curve in Weierstrass form, as described in [2, §8], or as implemented in Magma.

2. Proofs via equations for X_{E}(N, ε)

We prove Theorems 1.1 and 1.2 in the cases (N, ε) = (7, 3), (11, 1) and (12, 1). The case (N, ε) = (7, 3) was treated in [5, Section 8.2], but as this has not been published before, we include the details for completeness.

Case (N,ε) = (7,3). Let E be the elliptic curve y^{2} = x^{3} + ax + b. The following equation for X_{E}(7, 3), as a quartic curve in ℙ^{2}, was computed by Poonen, Schaefer and Stoll [20, Section 7.2], building on work of Halberstadt and Kraus [12].

F (a, b; x, y, z) = −a^{2}x^{4} + 2abx^{3}y − 12bx^{3}z − 6(a^{3} + 6b^{2})x^{2}y^{2} + 6ax^{2}z^{2}
+ 2a^{2}bxy^{3} − 12abxy^{2}z + 18bxyz^{2} + (3a^{4} + 19ab^{2})y^{4}
− 2(4a^{3} + 21b^{2})y^{3}z + 6a^{2}y^{2}z^{2} − 8ayz^{3} + 3z^{4}.

Replacing E by a quadratic twist does not change the isomorphism class of X_{E}(7, 3). This is borne out by the identity

F (λ^{2}a, λ^{3}b; λx, y, λ^{2}z) = λ^{8}F (a, b; x, y, z).

The surface Z(7, 3) is the quotient of {F = 0} ⊂ 𝔸^{2} × ℙ^{2} by this 𝔾_{m}-action. We have F (y, x_{1}y; x_{2}, T, y) = y^{2}(cy^{2} + hy − f) where

c = (T^{2} + 1)(3T^{2} − 8T + 3),
h = T^{3}(19T − 42)x_{1}^{2} + 2T (T − 3)^{2}x_{1}x_{2} − 6(T^{2} − 1)x_{2}^{2},
f = 36T^{2}x_{1}^{2}x_{2}^{2} − 2(T − 6)x_{1}x_{2}^{3} + x_{2}^{4}.

Therefore Z(7, 3) is birational to the total space for the genus one curve over ℚ(T) with equation Y^{2} = h^{2} + 4cf. This is a double cover of ℙ^{1} with a rational point above (x_{1} : x_{2}) = (1 : 0). Putting this elliptic curve in Weierstrass form, and replacing T by (6T − 3)/(4T + 4), gives the equation in the statement of Theorem 1.1.

Case (N,ε) = (11,1). Let E be the elliptic curve y^{2} = x^{3} + ax + b. Equations for X_{E}(11, 1) as a curve in ℙ^{4} were computed in [9, Theorem 1.2]. These equations are the 4 × 4 minors of the 5 × 5 Hessian matrix of the cubic form

F (a, b; v, w,x, y, z) = v^{3} + av^{2}z − 2avx^{2} + 4avxy − 6bvxz + avy^{2} + 6bvyz

+ a^{2}vz^{2} − w^{3} + aw^{2}z − 4awx^{2} − 12bwxz + a^{2}wz^{2} − 2bx^{3} + 3bx^{2}y

+ 2a^{2}x^{2}z + 6bxy^{2} + 4abxz^{2} + by^{3} − a^{2}y^{2}z + abyz^{2} + 2b^{2}z^{3}.

Replacing E by a quadratic twist does not change the isomorphism class of X_{E}(11, 1). This is borne out by the identity

F (λ^{2}a, λ^{3}b; λ^{2}v, λ^{2}w, λx, λy, z) = λ^{6}F (a, b; v, w, x, y, z).

We may describe Z(11, 1) as the quotient of a 3-fold in 𝔸^{2} × ℙ^{4} by this 𝔾_{m}-action.

We start by using the discriminant condition 4a^{3} + 27b^{2} ≠ 0 to simplify the equations for X_{E}(11, 1). The polynomials

F_{1} = vz + 2wz + x^{2} − xy − y^{2},
F_{2} = axz + bz^{2} − vx + vy − 2wx,
F_{3} = a^{2}z^{2} + 2awz − 4ax^{2} − 12bxz − 3w^{2},
F_{4} = a^{2}z^{2} + 2avz − 2ax^{2} + 4axy + ay^{2} − 6bxz + 6byz + 3v^{2},
F_{5} = 2a^{2}yz − abz^{2} − 4avx − 2avy − 6bvz − 3bx^{2} − 12bxy − 3by^{2},

are linear combinations of the derivatives of F, where the matrix implicit in taking these linear combinations is invertible if 4a^{3} + 27b^{2} ≠ 0. Now X_{E}(11, 1) is defined by the 4 × 4 minors of the 5 × 5 Jacobian matrix (M say) of F_{1}, …, F_{5}.

We make the substitutions

a = (4U + 3x_{3})x_{4} − 3x_{5}^{2},
b = x_{2}(x_{1} + x_{3})x_{4} − (4U + 3x_{3})x_{4}x_{5} + 2x_{5}^{3},
(v, w, x, y, z) = (x_{2}x_{4} + x_{4}x_{5} + x_{5}^{2}, x_{3}x_{4} − x_{5}^{2}, x_{5}, x_{4}, 1).

We have 4a^{3} + 27b^{2} = x_{4}h for some polynomial h. We add x_{5} times the first row of M to the second row. We then divide all but the first row by x_{4}. Let I ⊂ ℚ[U, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}] be the ideal generated by the 4 × 4 minors of M, and

J = {f ∈ ℚ[U, x_{1}, x_{2}, x_{3}, x_{4}, x_{5}] : x_{2}hf ∈ I}.

Using the Gröbner basis machinery in Magma we find that J ∩ ℚ[U, x_{1}, x_{2}, x_{3}, x_{4}] is generated by 3 homogeneous polynomials of degree 4. These define a surface in ℙ^{4} of degree 12. By the substitution

T = \frac{4(2x_{1} − x_{2} + x_{3})U + (x_{1}x_{2} − x_{2}^{2} + x_{3}x_{4})}{2(2x_{4} − x_{2})U + 2(x_{1}x_{2} − x_{2}^{2} + x_{3}x_{4})}

this surface is birational to the surface {Q_{1} = Q_{2} = 0} in 𝔸^{1} × ℙ^{3} where

Q_{1} = 4(T − 2)x_{1}x_{2} + 8x_{1}x_{3} + (T − 2)^{2}x_{2}^{2}
+ 4(T − 2)x_{2}x_{3} + 2(T^{2} − T + 1)x_{2}x_{4} + 4x_{3}^{2} − 4T x_{3}x_{4},
Q_{2} = 8x_{1}^{2} + 16x_{1}x_{3} − 4(2T − 1)x_{1}x_{4} − T (T − 2)x_{2}^{2} − T^{2}x_{2}x_{3}

− 2(T^{2} − T + 1)x_{2}x_{4} − 2(T − 4)x_{3}^{2} − 2(T^{2} + 4T − 1)x_{3}x_{4}.

These same equations define a genus one curve in ℙ^{3} defined over ℚ(T), with a rational point at (x_{1} : x_{2} : x_{3} : x_{4}) = (0 : 0 : 0 : 1). Putting this elliptic curve in Weierstrass form gives the equation in the statement of Theorem 1.2.

Case (N,ε) = (12,1). Let E be the elliptic curve y^{2} = x^{3} +ax+b. Equations for X_{E}(12, 1) as a curve in ℙ^{5} were computed in [5, Theorem 1.7.10]. These equations are F_{0} = F_{1} = F_{2} = F_{3} = 0 where

F_{0} = −X^{2}Z + aXY^{2} + 6bY^{3} − 6aY^{2}Z − 12Z^{3},
F_{1} = X^{2} + 12XZ + 36Z^{2} − 2u_{0}u_{2} − u_{1}^{2} + au_{2}^{2},
F_{2} = 4aXY + 36bY^{2} − 24aY Z − 2u_{0}u_{1} + 2au_{1}u_{2} + bu_{2}^{2},
F_{3} = 8aXZ − 4a^{2}Y^{2} − u_{0}^{2} + 2bu_{1}u_{2}.

These polynomials satisfy

F_{i}(λ^{2}a, λ^{3}b; λX, Y, λZ, λ^{2}u_{0}, λu_{1}, u_{2}) = λ^{m_{i}}F_{i}(a, b; X, Y, Z, u_{0}, u_{1}, u_{2})

where (m_{0}, m_{1}, m_{2}, m_{3}) = (3, 2, 3, 4). Again, it is our aim to quotient out by this 𝔾_{m}-action. We do this by setting (X + 6Z)Y = u_{2}^{2}. Specifically, we substitute (X, Y, Z, u_{0}, u_{1}, u_{2}) = (x^{2} − 6y, 1, y, vx, wx, x) and then solve for a and b so that the first two equations are satisfied. In the remaining two equations we substitute

v = 2(w − 2y)y + \frac{T + 1}{T − 1}(x^{2}(y + 1) − (w + 2y)^{2}).

The resultant with respect to w is f(T)x^{14}y^{2}g(x, y)^{2}h(x, ỹ) where f(T) is a rational function in T, g(x, y) = x^{6}(y + 1) − 9y^{2}(x^{2} + 4)^{2},

h(x, y) = (T + 1)^{2}x^{2}y^{2} + (T + 2)(T^{2} + 3)^{2}x^{2}

− 4(T − 1)(T + 3)^{2}xy + 4(T + 3)^{2}y^{2} + 12T (T + 1)^{2}(T + 3)^{2}

and ỹ = (144(T + 1)y + (T + 3)^{2}((T − 3)x^{2} + 12(T + 1)))/(8(T + 3)x). Therefore Z(12, 1) is birational to the total space for the genus one curve C = {h = 0} in 𝔸^{2} defined over ℚ(T). Replacing x by 2(T + 3)/(T^{2} + 3)x, and completing the square in y shows that C has equation

(1)  Y^{2} = −(T + 2)x^{4} − (4T^{3} + 5T^{2} + 6T + 9)x^{2} − 3T (T^{2} + 3)^{2}.

This gives a genus one fibration on Z(12, 1) defined over ℚ, but without a ℚrational section. Indeed the fibres with T > 0 have no real points.

We now find another genus one fibration that does have a ℚ-rational section. Let F (x_{1}, x_{2}, x_{3}) be the unique homogeneous polynomial of degree 6 with the property that F (x, T, 1) is the right hand side of (1). Then F is the discriminant of the following quadratic in T.

x_{1}^{2}x_{2} + (T^{2} + 2)x_{1}^{2}x_{3} + 2T x_{1}x_{2}^{2} − 2T x_{1}x_{2}x_{3} + T^{2}x_{2}^{3} + 3x_{2}^{2}x_{3} + 3T^{2}x_{2}x_{3}^{2} + 9x_{3}^{3} = 0

This same equation defines a genus one curve in ℙ^{2} defined over ℚ(T), with a rational point at (x_{1} : x_{2} : x_{3}) = (1 : 0 : 0). Putting this elliptic curve in Weierstrass form gives the equation in the statement of Theorem 1.1.

3. Degree 3 covers of K3-surfaces

We prove Theorems 1.1 and 1.2 in the cases (N, ε) = (6, 5), (10, 1) and (10, 3). In the first of these cases, Chen’s equations for X_{E}(6, 5) already give a genus one fibration on Z(6, 5), but one without a section. The content of Theorem 1.1 in this case is that we can find another genus one fibration that does have a section.

For N an odd integer, let Z^{∗}(N, ε) be the double cover of Z(N, ε) that parametrises pairs of elliptic curves whose ratio of discriminants is a square.

Theorem 3.1. If (N, ε) = (3, 2), (5, 1) or (5, 2) then Z^{∗}(N, ε) is a double cover of ℙ^{2}, ramified over the union of two cuspidal cubics, with equation

(2)  y^{2} = F_{+}(u, v, w)F_{−}(u, v, w)

where

Z^{∗}(3, 2) : F_{±} = u(u + 3v ± w)^{2} + 4v^{3},
Z^{∗}(5, 1) : F_{±} = u(u^{2} − 11uv − v^{2}) + w^{2}(12u + v) ± 2w(3u^{2} − 4uv + 4w^{2}),
Z^{∗}(5, 2) : F_{±} = u^{2}(11v + 8w) + w^{2}(8u − v + 4w) ± 2u(2v − w)(4u − v + 4w).

Proof. Let E be the elliptic curve y^{2} = x^{3} + ax + b. We put Δ = −4a^{3} − 27b^{2}, and define polynomials

f(x) = x^{3} + ax + b,
g(x) = 3ax^{4} + 18bx^{3} − 6a^{2}x^{2} − 6abx − a^{3} − 9b^{2},
h(x) = 3ax^{2} + 9bx − a^{2},
j(x) = 27bx^{3} − 18a^{2}x^{2} − 27abx − 2a^{3} − 27b^{2}.

If we assign the variables x, a, b weights 1, 2, 3, then each of these polynomials is homogeneous. We note that j^{2} = −4h^{3} − 27Δf^{2}. Case (N,ε) = (3,2). The following equations for the family of curves parametrised by X_{E}(3, 2) are taken from [7, Section 13]. Starting from the Klein form^{1}

D(ξ, η) = −27aξ^{4} − 54bξ^{3}η − 18a^{2}ξ^{2}η^{2} − 54abξη^{3} + (a^{3} − 27b^{2})η^{4},

we define

A(ξ, η) = \frac{1}{108} | \begin{matrix}D_{ξξ} & D_{ξη} \\ D_{ηξ} & D_{ηη} \end{matrix} | , and B(ξ, η) = \frac{1}{36} | \begin{matrix}D_{ξ} & D_{η} \\ A_{ξ} & A_{η} \end{matrix} | ,

where the subscripts denote partial derivatives. These forms satisfy the syzygy

(3)  −4A^{3} − 27B^{2} = 16(4a^{3} + 27b^{2})^{2}D^{3}.

1We obtain D from 𝔇(ξ, η) in [7, Section 9] by putting c_{4} = −48a, c_{6} = −864b, multiplying ξ by 12, and dividing through by 2^{12}3^{3}.

The family of elliptic curves 3-congruent to E with power ε = 2 is given by

y^{2} = x^{3} + A(ξ, η)x + B(ξ, η).

We dehomogenise by putting (ξ, η) = (x, 1). Then D = f_{x}^{3} − 27f^{2} = j − 3f_{x}h where f_{x} = 3x^{2} + a. The quantities (u, v, r, s) = (D, f_{x}h, h^{3}, 3^{6}Δf^{4}) are related by

(4)  (4r + (u + 3v)^{2})(ru − v^{3}) = rs.

As we verify in Remark 3.2 below, this is an equation for Z(3, 2) in ℙ(1, 1, 2, 3) where the coordinates u, v, r, s have weights 1, 1, 2, 3. We see by (3) that, up to squared factors, the ratio of discriminants is s/u. We substitute s = uw^{2} in (4) to give a quadratic in r whose discriminant is the polynomial F_{+}F_{−} in the statement of the theorem.

Case (N,ε) = (5,1). The following equations for the family of curves parametrised by X_{E}(5, 1) are taken from [7, Section 13]. Starting from the Klein form^{2}

D(λ, μ) = λ^{12} + 22aλ^{10}μ^{2} + 220bλ^{9}μ^{3} − 165a^{2}λ^{8}μ^{4} − 528abλ^{7}μ^{5}
− 220(a^{3} + 12b^{2})λ^{6}μ^{6} + 264a^{2}bλ^{5}μ^{7} − 165a(5a^{3} + 32b^{2})λ^{4}μ^{8}
− 880b(3a^{3} + 20b^{2})λ^{3}μ^{9} + 22a^{2}(25a^{3} + 168b^{2})λ^{2}μ^{10}
+ 20(19a^{4}b + 128ab^{3})λμ^{11} + (125a^{6} + 1792a^{3}b^{2} + 6400b^{4})μ^{12},

we define

A(λ, μ) = \frac{1}{5808} | \begin{matrix}D_{λλ} & D_{λμ} \\ D_{μλ} & D_{μμ} \end{matrix} | , and B(λ, μ) = \frac{1}{360} | \begin{matrix}D_{λ} & D_{μ} \\ A_{λ} & A_{μ} \end{matrix} | ,

where the subscripts denote partial derivatives. These forms satisfy the syzygy

(5)  4A^{3} + 27B^{2} = (4a^{3} + 27b^{2})D^{5}.

The family of elliptic curves 5-congruent to E with power ε = 1 is given by

y^{2} = x^{3} + A(λ, μ)x + B(λ, μ).

We dehomogenise by putting (λ, μ) = (x, 1). Then

D = 4kf − 3(f^{2} + g)^{2} + 32Δ(f^{2} + g),

where k(x) = f^{3} + fj + 4Δf + 3g(xf_{x} − 2f) = x^{9} + 12ax^{7} + 84bx^{6} + … The quantities (t, u, v, r, s) = (4f, 2(f^{2} + g), 16Δ, 4k, D) are related by

(6)  \begin{aligned}r^{2} + st^{2} & = u(u^{2} − 11uv − v^{2}) + (12u + v)s, \\ rt & = 3u^{2} − 4uv + 4s. \end{aligned}

2We obtain D from D(λ, μ) in [7, Section 8] by putting c_{4} = −48a, c_{6} = −864b, multiplying λ by 12, and dividing through by 2^{24}3^{12}.

These are equations for Z(5, 1) in ℙ(1, 2, 2, 3, 4) where the coordinates t, u, v, r, s have weights 1, 2, 2, 3, 4. We see by (5) that, up to squared factors, the ratio of discriminants is s. Putting s = w^{2} we obtain from (6) the equation

(r^{2} − st^{2})^{2} = (r^{2} + st^{2})^{2} − 4s(rt)^{2} = F_{+}(u, v, w)F_{−}(u, v, w)

where F_{±} are the polynomials in the statement of the theorem.

Case (N,ε) = (5,2). The following equations for the family of curves parametrised by X_{E}(5, 2) are taken from [8, Theorem 5.8]. Starting from the Klein form^{3}

D(λ, μ) = (125a^{3} − 432b^{2})λ^{12} + 2430a^{2}bλ^{11}μ − 22a(25a^{3} − 378b^{2})λ^{10}μ^{2}

− 110b(11a^{3} − 108b^{2})λ^{9}μ^{3} − 165a^{2}(5a^{3} − 27b^{2})λ^{8}μ^{4} − 132ab(53a^{3} − 189b^{2})λ^{7}μ^{5} + 220(a^{6} − 123a^{3}b^{2} + 81b^{4})λ^{6}μ^{6} + 132a^{2}b(19a^{3} − 297b^{2})λ^{5}μ^{7} − 165(a^{7} − 26a^{4}b^{2} + 189ab^{4})λ^{4}μ^{8} − 110(3a^{6}b − 34a^{3}b^{3} + 135b^{5})λ^{3}μ^{9} − 22a^{2}(a^{3} − 3b^{2})(a^{3} + 27b^{2})λ^{2}μ^{10} − 10ab(5a^{6} + 82a^{3}b^{2} + 189b^{4})λμ^{11} + (a^{9} − a^{6}b^{2} − 181a^{3}b^{4} − 675b^{6})μ^{12},

we define

A(λ, μ) = \frac{1}{1452} | \begin{matrix}D_{λλ} & D_{λμ} \\ D_{μλ} & D_{μμ} \end{matrix} | , and B(λ, μ) = \frac{−1}{180} | \begin{matrix}D_{λ} & D_{μ} \\ A_{λ} & A_{μ} \end{matrix} | ,

where the subscripts denote partial derivatives. These forms satisfy the syzygy

(7)  −4A^{3} − 27B^{2} = 16(4a^{3} + 27b^{2})^{2}D^{5}.

The family of elliptic curves 5-congruent to E with power ε = 2 is given by

y^{2} = x^{3} + A(λ, μ)x + B(λ, μ).

We dehomogenise by putting (λ, μ) = (x, 1). Then

(8)  D = 16Δf^{4} − g^{3} + 4(2g^{3} − g^{2}j − 4Δf^{2}g).

The quantities (r, s, v, w) = (Δf^{4}, Δf^{2}g, g^{3}, 2g^{3} − g^{2}j − 4Δf^{2}g) are related by

(9)  r(4s − 2v + w)^{2} + 27rsv + sw^{2} − s^{2}(v − 4w) = 0.

This is an equation for Z(5, 2) as a cubic surface in ℙ^{3}. We see from (7) and (8) that, up to squared factors, the ratio of discriminants is r(16r − v + 4w). Putting r(16r − v + 4w) = (4r − u)^{2}, where u is a new variable, and using this equation to eliminate r from (9), we obtain a quadratic in s whose discriminant is the polynomial F_{+}F_{−} in the statement of the theorem. □

3We obtain D from D(λ, μ) in [8, Section 5] by putting c_{4} = −48a, c_{6} = −864b, multiplying λ by 12, and dividing through by 2^{36}3^{15}.

Remark 3.2. Let (N, ε) = (3, 2), (5, 1) or (5, 2). We saw in the proof of Theorem 3.1 that one model for Z(N, ε) is the weighted projective plane ℙ(1, 2, 3) where the co-ordinates x, a, b have weights 1, 2, 3. We mapped this to another model for Z(N, ε) defined by (4), (6) or (9). The inverse maps are as follows.

(3, 2) \{ \begin{matrix}x & = r + v^{2}, \\ a & = −3r(r + uv + 2v^{2}), \\ b & = r(u + 3v)(ru + v^{3}) + 2r^{2}(r + 3v^{2}), \end{matrix}

(5, 1) \{ \begin{matrix}x & = 32u − v + 5t^{2}, \\ a & = −3(8u − v)(32u − v) − 288rt + 30(28u + v)t^{2} − 75t^{4}, \\ b & = −2(32u − v)^{2}(4u + v) − 144(32u − v)rt + 6(32u − v)(88u − 5v)t^{2} \\ & + 1008rt^{3} − 150(28u + v)t^{4} + 250t^{6}, \end{matrix}

(5, 2) \{ \begin{matrix}x & = 4rs + 4rv + rw − s^{2}, \\ a & = 3(8rs^{3} + 4rs^{2}v + 6rs^{2}w + rsw^{2} − s^{4}), \\ b & = r^{2}s(16s^{3} − 8s^{2}v − 24s^{2}w − 40svw − 15sw^{2} + 4vw^{2} − 2w^{3}) \\ & + rs^{3}(24s^{2} + 8sv + 34sw + 7w^{2}) − 2s^{6}. \end{matrix}

Remark 3.3. There are two naturally defined involutions on the K3-surfaces in Theorem 3.1. The first switches the sign of y, and corresponds to swapping over the pair of N-congruent elliptic curves. The second is given on Z^{∗}(3, 1) and Z^{∗}(5, 1) by switching the sign of w, and on Z^{∗}(5, 2) by (u, v, w, y) ↦ (ũ, v, w, (ũ/u)^{2}y) where ũ = u(v − 4w)/(8u − (v − 4w)). This second involution switches the choice of square root for the ratio of discriminants. The two involutions commute, and the second swaps over the curves F_{+} = 0 and F_{−} = 0.

Remark 3.4. For a suitable parametrisation of the cuspidal cubic F_{+} = 0, we obtain a family of elliptic curves with j-invariant

(3, 1) :  j = 27(T − 3)^{3}(T + 1)^{3}/T^{3},
(5, 1) :  j = (T + 5)^{3}(T^{2} − 5)^{3}(T^{2} + 5T + 10)^{3}/(T^{2} + 5T + 5)^{5},
(5, 2) :  j = 125T (2T + 1)^{3}(2T^{2} + 7T + 8)^{3}/(T^{2} + T − 1)^{5}.

These correspond to X_{s}^{+}(3), X_{s}^{+}(5) and X_{ns}^{+}(5), where X_{s}^{+}(N) and X_{ns}^{+}(N) are the modular curves associated to the normaliser of a split or non-split Cartan subgroup of level N. We may compute X_{s}^{+}(N) as the quotient of X_{0}(N^{2}) by the Fricke involution, whereas the formula for X_{ns}^{+}(5) is taken from [4, Corollary 5.3]. The use of these modular curves to construct pairs of N-congruent elliptic curves is described further in [11].

Let N be an odd integer and let ε ∈ (ℤ/2Nℤ)^{×}. Then X_{E}(2N, ε) → X_{E}(N, ε) is geometrically a Galois covering with Galois group PSL_{2}(ℤ/2ℤ) ≅ S_{3}. Since elliptic curves which are 2-congruent have ratio of discriminants a square, it follows that Z(2N, ε) → Z^{∗}(N, ε) is a degree 3 cover. In the cases (2N, ε) = (6, 5), (10, 1) and (10, 3) the surface Z(2N, ε) has an elliptic fibration. The pushfoward of a fibre gives a divisor class D on the K3-surface Z^{∗}(N, ε) with D^{2} = 2. Using this divisor class D we may write Z^{∗}(N, ε) as a double cover of ℙ^{2}. We have arranged (with the benefit of hindsight) that the equations in Theorem 3.1 write Z^{∗}(N, ε) as a double cover of ℙ^{2} in exactly this way.

The equations for Z(2N, ε) in Theorems 1.1 and 1.2 may be obtained from the equations for Z^{∗}(N, ε) in Theorem 3.1 as follows. The tangent line to a general point on the cuspidal cubic F_{+}(u, v, w) = 0 has equation:

(10)  (2N, ε) = (6, 5)  (T^{3} − 1)u + 3(T − 1)v − w = 0,
(11)  (2N, ε) = (10, 1)  (T − 2)u − T (T − 1)^{2}v + 2(T − 1)w = 0,
(12)  (2N, ε) = (10, 3)  T^{3}u − (T + 1)v − T^{2}w = 0.

We parametrise this line, and substitute into the right hand side of the equation y^{2} = F_{+}(u, v, w)F_{−}(u, v, w). After cancelling a squared factor (which arises since we chose a tangent line) the right hand side is a binary quartic with a linear factor. We now have the equation for a genus one curve over ℚ(T) with a rational point. Putting this elliptic curve in Weierstrass form gives the equations for Z(6, 5), Z(10, 1) and Z(10, 3) in Theorems 1.1 and 1.2.

It remains to show that these degree 3 covers of Z^{∗}(N, ε) are the ones we wanted. We use the following lemma.

Lemma 3.5. Let K be a field of characteristic not 2 or 3. Elliptic curves E_{1} and E_{2} over K with j-invariants j_{1} and j_{2}, with j_{1}, j_{2} ∉ {0, 1728}, are 2-congruent if and only if there exist m, x ∈ K satisfying (j_{1} − 1728)(j_{2} − 1728) = m^{2} and

x^{3} − 3j_{1}j_{2}x − 2j_{1}j_{2}(m + 1728) = 0.

Proof. This follows from the formulae in [23] or [7, Sections 8 and 13] by a generic calculation. □

We illustrate the use of Lemma 3.5 in the case (2N, ε) = (10, 3), the other cases being similar. Above each point (u : v : w) ∈ ℙ^{2} there are a pair of points on Z^{∗}(5, 2) possibly defined over a quadratic extension. These points correspond to a pair of elliptic curves, say with j-invariants j_{1} and j_{2}. A calculation using the formulae in Remark 3.2 shows that, for m a suitable choice of square root of (j_{1} − 1728)(j_{2} − 1728), we have

j_{1}j_{2} = G_{6}(u, v, w)H(u, v, w)^{2}
j_{1}j_{2}(m + 1728) = G_{9}(u, v, w)H(u, v, w)^{3}

where

G_{6}(u, v, w) = 640u^{4}v^{2} − 768u^{4}vw − 72u^{3}v^{3} − 240u^{3}v^{2}w + …
G_{9}(u, v, w) = 6912u^{7}v^{2} − 1376u^{6}v^{3} − 14976u^{6}v^{2}w + …

are irreducible homogeneous polynomials of degrees 6 and 9, and H ∈ ℚ(u, v, w) is a rational function. Finally we claim that the polynomials

(13)  X^{3} − 3G_{6}(u, v, w)X − 2G_{9}(u, v, w) = 0,

arising from Lemma 3.5, and

(14)  uT^{3} − wT^{2} − vT − v = 0,

appearing in (12), define the same cubic extension. Indeed we find by computer algebra that if (14) has root T_{0} then (13) has root

X_{0} = 3u^{2}(8u − 3v − 4w)T_{0}^{2} + 12u(2uv − 4uw + vw)T_{0}
− 16u^{2}v + 6uv^{2} + 8uw^{2} − vw^{2} + 4w^{3}.

4. Computing the Picard numbers

Let E/ℚ(T) be one of the elliptic curves in Theorems 1.1 and 1.2. We write X → ℙ^{1} for the minimal fibred surface with generic fibre E. The reduction of E mod p is an elliptic curve E_{p}/𝔽_{p}(T), and the reduction of X mod p is a surface X_{p}/𝔽_{p}. We will always take p to be a prime of good reduction.

Let \overline{X} = X ×_{ℚ} \overline{ℚ} and \overline{X}_{p} = X_{p} ×_{𝔽_{p}} \overline{𝔽}_{p}. The Shioda-Tate formula [24, Corollary 5.3] tells us that

(15)  rankE(\overline{ℚ}(T)) + 2 + ∑_{t∈ℙ^{1}(\overline{ℚ})}(m_{t} − 1) = rank NS(\overline{X}),

and

(16)  rankE_{p}(\overline{𝔽}_{p}(T)) + 2 + ∑_{t∈ℙ^{1}(\overline{𝔽}_{p})}(m_{t} − 1) = rank NS(\overline{X}_{p}),

where m_{t} is the number of irreducible components in the fibre above t. We write ρ and ρ_{p} for the numbers on the right of (15) and (16). These are the geometric Picard numbers of X and X_{p}. The sections exhibited in Tables 3 and 4 give a lower bound for rankE(\overline{ℚ}(T)) and hence by (15) a lower bound for ρ. These lower bounds are exactly the values recorded in Tables 1 and 2.

Let X → ℙ^{1} be a minimal elliptic surface with non-constant j-invariant, and let m = χ(O_{X}). This may be computed from the fact that sum of the Euler numbers of the singular fibres is 12m. By [17, Lemma IV.1.1] the Hodge diamond of X is

\begin{aligned}& & h^{0,0} \\ & h^{1,0} & & h^{0,1} \\ h^{2,0} & & h^{1,1} & & h^{0,2} \\ & h^{2,1} & & h^{1,2} \\ & & h^{2,2} \end{aligned}  \begin{aligned}& & 1 \\ & 0 & & 0 \\ m − 1 & & 10m & & m − 1 \\ & 0 & & 0 \\ & & 1 \end{aligned}

The surfaces in Theorem 1.1 have m = 2 and those in Theorem 1.2 have m = 3. To tie in with [14, Theorem 4], we note that p_{g} = h^{2,0} = m − 1. By the Lefschetz theorem on (1, 1)-classes we have ρ ≤ h^{1,1} = 10m. This already determines ρ in all cases with N ≤ 8. It remains for us to improve this upper bound by 1 in the cases (N, ε) = (9, 1), (12, 1), (9, 2), and to improve it by 2 in the cases (N, ε) = (10, 1), (10, 3), (11, 1).

The main tool we wish to use (see [26, Proposition 6.2]) is that there is an injective map NS(\overline{X}) → NS(\overline{X}_{p}) that preserves the intersection pairing.

Let f_{p}(x) be the characteristic polynomial of Frobenius acting on H_{ét}^{2}(\overline{X}_{p}, ℚ_{ℓ}(1)), normalised so that f_{p}(0) = 1. This is a polynomial of degree b_{2} = 12m − 2, independent of the choice of prime ℓ ≠ p. By the Weil conjectures it satisfies the functional equation f_{p}(x) = ±x^{b_{2}}f_{p}(1/x). The polynomials f_{p}(x) may be computed from the numbers n_{r} = |X_{p}(𝔽_{p^{r}})| using the Lefschetz trace formula. See for example [27, Section 3], where f_{p} is denoted f̃_{p}. We used both the functional equation and the Magma function FrobeniusActionOnTrivialLattice to limit how many n_{r} we had to compute. The polynomials f_{p}(x) for two carefully chosen primes of good reduction are recorded in Table 5.

Let Δ_{p} ∈ ℚ^{×}/(ℚ^{×})^{2} be the absolute value of the determinant of the intersection pairing on NS(\overline{X}_{p}). It may be computed using either of the following two lemmas.

Lemma 4.1. Write f_{p}(x) = g_{p}(x)h_{p}(x) where every root of g_{p} is a root of unity, and no root of h_{p} is a root of unity. Then ρ_{p} ≤ deg g_{p} and in the case of equality we have Δ_{p} ≡ h_{p}(1)h_{p}(−1) mod (ℚ^{×})^{2}.

Proof. The first part is described for example in [27]. The Tate conjecture predicts that this inequality is always an equality, and this has been proved in many cases. See [13, Section 17.3] for the history of this problem and further references. The formula for Δ_{p} is a small refinement of a result of Kloosterman, that in turn depends on known cases of the Artin-Tate conjecture.

Let F_{k}(x) = ∏(1−p^{k}α_{i}^{k}x) where f_{p}(x) = ∏(1−α_{i}x). The result of Kloosterman [15, Proposition 4.7], is that if k is an even integer with α_{i}^{k} = 1 for all α_{i} which

Table 5

(N, ε)  Characteristic polynomial of Frobenius
(9, 1) f_{5}(x) = (x − 1)^{16}(x + 1)^{2}(x^{2} + x + 1)(x^{2} + \frac{7}{5}x + 1)
f_{7}(x) = (x − 1)^{18}(x + 1)^{2}(x^{2} + \frac{10}{7}x + 1)
(12, 1) f_{5}(x) = (x − 1)^{16}(x + 1)^{4}(x^{2} + \frac{6}{5}x + 1)
f_{11}(x) = (x − 1)^{12}(x + 1)^{8}(x^{2} + \frac{6}{11}x + 1)
(9, 2) f_{7}(x) = (x − 1)^{24}(x + 1)^{2}(x^{2} + x + 1)^{2}(x^{2} + \frac{10}{7}x + 1)(x^{2} + \frac{13}{7}x + 1)
f_{13}(x) = (x − 1)^{24}(x^{2} + x + 1)^{3}(x^{2} + \frac{1}{13}x + 1)(x^{2} + \frac{25}{13}x + 1)
(10, 1) f_{7}(x) = (x − 1)^{24}(x + 1)^{2}(x^{2} + x + 1)^{2}(x^{2} + \frac{10}{7}x + 1)^{2}
f_{17}(x) = −(x − 1)^{25}(x + 1)^{5}(x^{2} − \frac{2}{17}x + 1)(x^{2} + \frac{25}{17}x + 1)
(10, 3) f_{31}(x) = (x − 1)^{24}(x + 1)^{2}(x^{2} + x + 1)^{2}(x^{2} + \frac{46}{31}x + 1)(x^{2} + \frac{58}{31}x + 1)
f_{37}(x) = (x − 1)^{28}(x + 1)^{2}(x^{2} + \frac{70}{37}x + 1)^{2}
(11, 1) f_{23}(x) = (x − 1)^{28}(x + 1)^{2}(x^{2} + \frac{42}{23}x + 1)(x^{2} + \frac{45}{23}x + 1)
f_{53}(x) = (x − 1)^{28}(x^{2} + x + 1)(x^{2} + \frac{25}{53}x + 1)(x^{2} + \frac{70}{53}x + 1)

are roots of unity, then

(17)  Δ_{p} = lim_{s→1} \frac{F_{k}(p^{−ks})}{(1 − p^{k(1−s)})^{ρ}} .

Let H_{k}(x) = ∏(1 − p^{k}β_{i}^{k}x) where h_{p}(x) = ∏(1 − β_{i}x). Then F_{k}(x) = (1 − p^{k}x)^{ρ}H_{k}(x), and (17) becomes

Δ_{p} = H_{k}(p^{−k}) = ∏_{i}(β_{i}^{k} − 1) = h_{p}(1)h_{p}(−1) ∏_{\begin{matrix}d|k \\ d>2 \end{matrix}} ∏_{i} Φ_{d}(β_{i})

where Φ_{d} is the dth cyclotomic polynomial. For d > 2 we claim that ∏_{i} Φ_{d}(β_{i}) is a rational square. By the functional equation we have

β_{1}, …, β_{2m} = γ_{1}, …, γ_{m}, γ_{1}^{−1}, …, γ_{m}^{−1}.

Since d > 2 we have Φ_{d}(x) = x^{ϕ(d)}Φ_{d}(1/x) where ϕ(d) is even, say ϕ(d) = 2n. Therefore γ_{i}^{−n}Φ_{d}(γ_{i}) = γ_{i}^{n}Φ_{d}(γ_{i}^{−1}). It follows that ∏_{i=1}^{m} γ_{i}^{−n}Φ_{d}(γ_{i}) ∈ ℚ and

∏_{i=1}^{2m} Φ_{d}(β_{i}) = ∏_{i=1}^{2m} β_{i}^{−n}Φ_{d}(β_{i}) = ( ∏_{i=1}^{m} γ_{i}^{−n}Φ_{d}(γ_{i}))^{2}.  □

Lemma 4.2. Suppose that P_{1}, … P_{r} generate a finite index subgroup of E_{p}(\overline{𝔽}_{p}(T)). Then we have Δ_{p} ≡ (∏_{t} c_{t}) Reg(P_{1}, …, P_{r}) mod (ℚ^{×})^{2} where the product is over t ∈ ℙ^{1}(\overline{𝔽}_{p}) and c_{t} is the number of irreducible components of multiplicity one in the fibre of \overline{X}_{p} above t.

Proof. See [24, Theorem 8.7 and (7.8)]. □

In the calculations below, we sometimes needed to find explicit generators for E_{p}(𝔽_{p}(T)). These were found by searching on 2-coverings, computed using 2descent in Magma, as implemented in the function field case by S. Donnelly.

If ρ = ρ_{p} = ρ_{q} for distinct primes p and q, then we have Δ_{p} ≡ Δ_{q} mod (ℚ^{×})^{2}. As observed by van Luijk [27], this can sometimes be used to improve our upper bound on ρ by 1. This is particularly useful since (assuming the Tate conjecture) ρ_{p} is always even. Indeed ρ_{p} = deg g_{p} = b_{2} − deg h_{p}, and deg h_{p} is even by the functional equation. See [15] and [18] for further examples.

Case (N,ε) = (9,1). We already know that ρ = 19 or 20. Since the Tate conjecture has been proved for elliptic K3-surfaces, equality holds in Lemma 4.1. By Lemma 4.1 we compute Δ_{5} = 3 · 17 and Δ_{7} = 2 · 3. Since these are different, it follows by the method of van Luijk that ρ = 19.

Case (N,ε) = (12,1). This is identical to the previous example, except that now Δ_{5} = 1 and Δ_{11} = 7.

Case (N,ε) = (9,2). We already know that ρ = 29 or 30. Let p = 7 or 13. By Lemma 4.1 and (16) we have rank E_{p}(\overline{𝔽}_{p}(T)) ≤ 3. We prove equality by exhibiting three independent points of infinite order in E_{p}(𝔽_{p}(T)). In addition to the reductions of the two points in Table 3, we have when p = 7 the point

(5T^{5} + 6T^{4} + 4T^{2} + 6T, T^{7} + 3T^{6} + 6T^{3} + 2T^{2} + 2T),

and when p = 13 the point

(4T^{6} +8T^{5} +3T^{4} +7T^{3} +5T^{2} +10T +2, 10T^{9} +4T^{8} +5T^{6} +5T^{5} +12T^{3} +4T^{2} +12).

Using either Lemma 4.1 or Lemma 4.2 we find that Δ_{7} = 2 and Δ_{13} = 17. Since these are different, it follows that ρ = 29.

In the cases (N, ε) = (10, 1), (10, 3), (11, 1) we aim to show that ρ = 28. We were unable^{4} to find a prime p with ρ_{p} = 28, despite computing the polynomials f_{p}(x) for all p < 150. This prompted us to try a variant of van Luijk’s method, were we use the intersection pairing to improve our upper bound for ρ by 2.

Case (N,ε) = (10,1). We already know that ρ = 28, 29 or 30. In addition to the point P_{1} = (0, 0) in Table 3 we have when p = 7 the points

Q_{1} = (6T^{6} + 6T^{4} + 4T^{3} + 5T^{2}, 4T^{9} + 6T^{8} + 6T^{7} + T^{6} + T^{5} + 3T^{4}),
Q_{2} = (T^{6} + 5T^{5} + 6T^{4} + 4T^{3} + 5T^{2}, 2T^{9} + 6T^{8} + 2T^{7} + T^{6} + 3T^{4}),

4There is presumably a systematic reason for this, similar to that described in [3].

and when p = 17 the points

R_{1} = (16T^{6} + 13T^{5} + 6T^{4} + 4T^{3} + 12T^{2}, 4T^{9} + 2T^{8} + 5T^{7} + 8T^{5} + 15T^{4}),
R_{2} = ((6T^{8} + 8T^{7} + 2T^{6} + 5T^{5} + 8T^{4} + 4T^{3} + T^{2})/(T + 6)^{2}, …).

Using either Lemma 4.1 or Lemma 4.2 we find that Δ_{7} = 1 and Δ_{17} = 2·59. Since these are different, it follows that ρ ≤ 29.

Reducing mod 7 or 17 does not change the Kodaira symbols of the singular fibres. So by Lemma 4.2 it will be enough for us to work with the height pairing on the Mordell-Weil group, rather than the intersection pairing on the full Néron-Severi group. We compute

Reg(P_{1}, uQ_{1} + vQ_{2}) = \frac{2}{75}(7u^{2} − 12uv + 18v^{2}),
Reg(P_{1}, xR_{1} + yR_{2}) = \frac{1}{450}(139x^{2} + 76xy + 316y^{2}).

If ρ = 29 then the equation \frac{2}{75}(7u^{2} − 12uv + 18v^{2}) = \frac{1}{450}(139x^{2} + 76xy + 316y^{2}) has a solution in rational numbers u, v, x, y not all zero. However this quadratic form of rank 4 is not locally soluble over the 3-adics. Therefore ρ = 28.

Case (N,ε) = (10,3). We already know that ρ = 28, 29 or 30. Let p = 31 or 37. Since p ≡ 1 (mod 3) the reductions of the points in Tables 3 and 4 give us points P_{1}, P_{2}, P_{3}, P_{4} ∈ E_{p}(𝔽_{p}(T)). In addition when p = 31 we have

Q_{1} = (20T^{4} + 13T^{3} + 30T^{2} + 6T, 5T^{5} + 4T^{4} + 29T^{3} + 4T^{2} + 5T),
Q_{2} = (7T^{6} + 12T^{5} + 9T^{4} + 19T^{2} + 13T + 4)/(T + 29)^{2}, …),

and when p = 37 we have

R_{1} = (36T^{4} + 11T^{3} + 4T^{2}, 26T^{5} + 34T^{4} + 2T^{3} + 15T^{2}),
R_{2} = (6T^{4} + 5T^{3} + T^{2} + 26T + 32, 29T^{5} + 35T^{4} + 2T^{3} + 15T^{2} + 19T + 10).

Using either Lemma 4.1 or Lemma 4.2 we find that Δ_{31} = 2 · 5 and Δ_{37} = 1. Therefore ρ ≤ 29.

As in the previous example, reducing mod 31 or 37 does not change the singular fibres. We compute

Reg(P_{1}, P_{2}, P_{3}, P_{4}, uQ_{1} + vQ_{2}) = \frac{5}{96}(25u^{2} − 4uv + 52v^{2}),
Reg(P_{1}, P_{2}, P_{3}, P_{4}, xR_{1} + yR_{2}) = \frac{1}{8}(5x^{2} + 8y^{2}).

If ρ = 29 then the equation \frac{5}{96}(25u^{2} − 4uv + 52v^{2}) = \frac{1}{8}(5x^{2} + 8y^{2}) has a solution in rational numbers u, v, x, y not all zero. However this quadratic form is not locally soluble over the 3-adics. Therefore ρ = 28.

Case (N,ε) = (11,1). We already know that ρ = 28, 29 or 30. Let P_{1} and P_{2} be the reductions mod p of the points in Table 3. In addition, when p = 23 we have

Q_{1} = (16T^{2} + 5T + 5, 21T^{3} + 15T^{2} + 3T + 18),
Q_{2} = (18T^{6} + 5T^{5} + 5T^{4} + 22T^{3} + 9T^{2})/(T + 16)^{2}, …),

and when p = 53 we have

R_{1} = (28T^{5} + T^{4} + 23T^{3} + 40T^{2} + 15T, …),
R_{2} = (49T^{6} + 44T^{5} + 38T^{4})/(T^{2} + 42T + 5)^{2}, …).

Using either Lemma 4.1 or Lemma 4.2 we find that Δ_{23} = 2 · 7 · 11 · 13 and Δ_{53} = 11 · 131. Therefore ρ ≤ 29.

Again, reducing mod 23 or 53 does not change the singular fibres. We compute

Reg(P_{1}, P_{2}, uQ_{1} + vQ_{2}) = \frac{11}{480}(57u^{2} − 46uv + 137v^{2}),
Reg(P_{1}, P_{2}, xR_{1} + yR_{2}) = \frac{1}{240}(541x^{2} − 228xy + 1196y^{2}).

If ρ = 29 then the equation \frac{11}{480}(57u^{2}−46uv+137v^{2}) = \frac{1}{240}(541x^{2}−228xy+1196y^{2}) has a solution in rational numbers u, v, x, y not all zero. However this quadratic form is not locally soluble over the 11-adics. Therefore ρ = 28.

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University of Cambridge, DPMMS, Centre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK

E-mail address: T.A.Fisher@dpmms.cam.ac.uk